Triple deck theory

Middle deck

The solution in the middle deck is found to be

${\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\sqrt {2}}f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)+\varepsilon A(\chi )f_{B}'\left({\frac {\eta }{\sqrt {2}}}\right)+\varepsilon ^{2}\Phi (\chi ,\eta )+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots }$

where ${\displaystyle A(\chi )}$ is referred to as the displacement function and ${\displaystyle P(\chi )}$ is referred to as the pressure function, to be determined from the upper and lower deck problems. Note that the correction to the Blasius stream function is of the order ${\displaystyle \varepsilon }$, although the pressure perturbation is only order ${\displaystyle \varepsilon ^{2}.}$

Upper deck

In the upper deck, the solution is found to given by

${\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\frac {\zeta }{\varepsilon }}-{\sqrt {2}}\beta -{\frac {\varepsilon }{\pi }}\int _{-\infty }^{+\infty }A'(\chi )\left[\tan ^{-1}\left({\frac {\chi -{\hat {\chi }}}{\zeta }}\right)+{\frac {\pi }{2}}\right]d{\hat {\chi }}+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots }$

where ${\displaystyle \beta =1.2168}$. Furthermore, the upper deck problem also provides the relation between the displacement and the pressure function as

${\displaystyle P(\chi )=\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {A'({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}\quad {\text{and}}\quad A'(\chi )=-\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {P({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}.}$

in which ${\displaystyle \mathrm {p.v.} }$ stands for Cauchy principal value. One may notice that the pressure function and the derivative of the displacement function (aka transverse velocity) forms a Hilbert transform pair (see Kramers–Kronig relations).

Lower deck

In the lower deck, the solution is given by

${\displaystyle {\frac {\psi }{\varepsilon ^{4}}}=\varepsilon ^{2}\Psi (\chi ,\xi )+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots }$

where ${\displaystyle \Psi (\chi ,\xi )}$ will satisfy a boundary-layer type equations driven by the pressure gradient ${\displaystyle dP/d\chi }$ and the slip-velocity of order ${\displaystyle \varepsilon ^{2}}$ generated by the middle deck. It is convenient to introduce ${\displaystyle {\hat {u}}=\partial \Psi /\partial \xi }$ and ${\displaystyle {\hat {v}}=-\partial \Psi /\partial \chi }$, where ${\displaystyle {\hat {u}}}$ and ${\displaystyle {\hat {v}}}$ must satisfy

${\displaystyle {\begin{aligned}{\frac {\partial {\hat {u}}}{\partial \chi }}+{\frac {\partial {\hat {v}}}{\partial \xi }}&=0,\\{\hat {u}}{\frac {\partial {\hat {u}}}{\partial \chi }}+{\hat {v}}{\frac {\partial {\hat {u}}}{\partial \xi }}&=-\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {A''({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}+{\frac {\partial ^{2}{\hat {u}}}{\partial \xi ^{2}}}.\end{aligned}}}$

These equations are subjected to the conditions

${\displaystyle \xi =0:\,\,\,{\begin{cases}{\hat {u}}={\hat {v}}=0,\,\,\chi \leq 0,\\{\frac {\partial {\hat {u}}}{\partial \xi }}={\hat {v}}=0,\,\,\chi >0,\end{cases}}\quad {\text{and}}\quad \xi \to \infty$ :{\hat {u}}\to {\frac {\xi +A(\chi )}{\sqrt {2\alpha ^{3}}}},}

${\displaystyle \chi \to -\infty$ :\,\,\,A\to 0,\quad {\text{and}}\quad \chi \to +\infty :\,\,\,A\to {\frac {\mu }{\lambda }}\chi ^{1/3}}

where ${\displaystyle \alpha =1.6552}$. The displacement function ${\displaystyle A(\chi )}$ and therefore ${\displaystyle P(\chi )}$ must be obtained as part of the solution. The above set of equations may resemble normal boundary-layer equations, however it has an elliptic character since the pressure gradient term now is non-local, i.e., the pressure gradient at a location ${\displaystyle \chi }$ depends on other locations as well. Because of this, these equations are sometimes referred to as the interactive boundary-layer equations. The numerical solution of these equations were obtained by Jobe and Burggraf in 1974.